We show how to perform matrix operations involving complex numbers in Excel.
We represent an m × n matrix of complex numbers by a m × 2n range in Excel. This range consists of an m × n range on the left and another m × n range on the right. The first of these ranges represents the real parts of the values of the complex numbers in the complex number matrix and the second of these ranges represents the imaginary parts of the complex numbers in the complex number matrix.
For example, the complex number matrix
can be represented by the Excel range B13:E14 shown in Figure 1
Figure 1 – Complex matrix in Excel
The left side in yellow (range B13:C14) contains the real values and the right side in green (range D13:E14) contains the imaginary values.
Essentially, we are expressing a complex matrix as A + Bi where A and B are matrices which only have real values.
Matrix addition, subtraction, multiplication and inverse on complex matrices are calculated using only real matrix operations, as follows:
Real Statistics Function: The Real Statistics Resource Pack supplies the following array functions, where Y and Z are ranges that represent complex matrices, while z is a range that represents a complex (scalar) number and k is a positive integer.
| ZAdd(Y, Z) = Y + Z | ZSub(Y, Z) = Y – Z | ZMult(Y, Z) = Y × Z |
| ZInverse(Z) = Z-1 | ZTranspose(Z) = ZT | ZIdentity(k) = identity |
| ZMultScalar(Y, z) = z × Y |
Note that ZIdentity(k) outputs a k × k identity matrix. If k is omitted it defaults to the number of rows in the highlighted range.
In Figure 1 and 2, we show the results of various complex matrix operations.
Figure 2 – Complex Matrix Operations – part 1
Figure 3 – Complex Matrix Operations – part 2
